MULTI-AGENT SOFT CONSTRAINT AGGREGATION

A Sequential Approach

Giorgio Dalla Pozza, Francesca Rossi and K. Brent Venable

Dept. of Pure and Applied Mathematics, University of Padova, Padova, Italy

Keywords:

Group decision making, Constraint satisfaction, Fuzzy systems.

Abstract:

We consider a scenario where several agents express their preferences over a common set of variable assign-

ments, by means of a soft constraint problem for each agent, and we propose a procedure to compute a variable

assignment which satisﬁes the agents’ preferences at best. Such a procedure considers one variable at a time

and, at each step, asks all agents to express its preferences over the domain of that variable. Based on such

preferences, a voting rule is used to decide on which value is the best for that variable. At the end, the val-

ues chosen constitute the returned variable assignment. We study several properties of this procedure and we

show that the use of soft constraints allows for a great ﬂexibility on the preferences of the agents, compared

to similar work in setting where agents model their preferences via CP-nets, where several restrictions on the

agents’ preferences need to be imposed to obtain similar properties.

1 INTRODUCTION

We consider scenarios in which a set of agents express

their preferences over a common set of objects, and

the aim is to choose one object which best satisﬁes

the preferences of the agents. We also assume that the

set of objects has a combinatorial structure. More pre-

cisely, each object is modelled as an elements of the

Cartesian product of a set of variable domains. This

is often the case in real life, since usually we express

preferenecs over objects that are characterized by a

set of features (the variables), each of which has some

possible instances (the variable domain).

Finally, we also assume that the agents express

their preferences over the objects in a compact way.

There are many formalisms suitable to do this; in this

paper we focus on soft constraints (Meseguer et al.,

2005). Thus each agent speciﬁes a set of soft con-

straints over the variables. Another well-known for-

malism to do this is CP-nets (Boutilier et al., 2004).

The aim of such formalisms to allow one to express in

time and space polynomial in the number of variables

an ordering over the et of all objects, which may be

exponential in such a number.

To deﬁne a procedure to aggregate the preferences

of such agents, we consider voting theory (Arrow and

amd K. Suzumura, 2002), a very wide research area

between economy theory and operation research, that

deals with elections, where voters (that we would call

agents) vote by expressing their preferences over a set

of candidates (that we would call objects), and a vot-

ing rule decides who the winner candidate is. Voting

theory provides many rules to aggregate preferences,

that take in input (a part of) the preference orderings

of the agents and gives as output the ”winner” object,

that is, the object that is considered to be the best ac-

cording to the rule.

The most naive way to use voting rules in our con-

text consists of choosing a voting rule and giving to it

what it needs to know about the preference orderings

of the agents, then running the voting rule and see

what result comes out. This is however not feasible

in general. In fact, if the chosen voting rule needs to

know a large part of the preference ordering from the

agents, it may take exponential time only to give the

input to the rule. A valid alternative is to use the vot-

ing rule several times, on each feature of the object

set. This approach is certainly more attractive com-

putationally, since usually the number of instances of

each feature is small.

What can be done in this situation is to study

when, even in presence of dependencies among fea-

tures, voting can be performed on each single feature

at a time rather than on complete objects. We there-

fore deﬁne a sequential voting procedure that, at each

step, applies one voting rule to the preferences of the

agents over the domain of a single variable. We then

study the properties of this sequential voting proce-

dure. In particular, we consider properties such as

Condorcet consistency, anynomity, neutrality, consis-

277

Dalla Pozza G., Rossi F. and Brent Venable K..

MULTI-AGENT SOFT CONSTRAINT AGGREGATION - A Sequential Approach.

DOI: 10.5220/0003156602770282

In Proceedings of the 3rd International Conference on Agents and Artiﬁcial Intelligence (ICAART-2011), pages 277-282

ISBN: 978-989-8425-40-9

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

tency, participation, efﬁciency, and monotonicity, and

we relate their presence to the corresponding proper-

ties of the voting rules used at each step of the proce-

dure.

This study has been done already for CP-nets

(Lang and Xia, 2009), showing that a sequential

single-feature voting protocol can ﬁnd a winner ob-

ject in polynomial time, and have several other desir-

able properties, when the CP-nets satisfy certain con-

ditions on their dependencies. We show that the use

of soft constraints allows us to avoid imposing many

restrictions on the preferences of the agents. In fact,

contrarily to CP-nets, soft constraints are not direc-

tional, and thus information can ﬂow from one vari-

able of a constraint to another one without a prede-

ﬁned ordering between them. This allows us to not

tie the variable ordering used by the sequential proce-

dure to the topology of the constraint graph of each

agent. This makes the approach much more generally

applicable. In fact, the tractability assumption over

the constraint graphs is similar to the assumptions that

CP-nets are acyclic. However, we do not need to im-

pose that the constraint graphs are compatible among

them and with a graph structure based on the variable

ordering.

2 BACKGROUND

Soft Constraints. A soft constraint (Meseguer

et al., 2005) involves a set of variables and associates

a value from a (totally or partially ordered) set to each

instantiation of its variables. Such a value is taken

from a c-semiring, which is deﬁned by hA,+,×,0, 1i,

where A is the set of preference values, + is a com-

mutative, associative, and idempotent operator, × is

used to combine preference values and is associative,

commutative, and distributes over +, 0 is the worst

element, and 1 is the best element. A c-semiring S

induces a partial or total order ≤

over the preference

values, where a ≤

b iff a+ b = b. A Soft Constraint

Satisfaction Problem (SCSP) is a tuple hV,D,C, Ai

where V is a set of variables, D is the domain of the

variables and C is a set of soft constraints over V as-

sociating values from c-semiring A.

An instance of the SCSP framework is obtained

by choosing a c-semiring. For instance, in classical

constraints we want all constraints to be satisﬁes, thus

we may choose the semiring S

CSP

= h{ false, true},

∨,∧, false,truei. If instead we want to maximize

the minimum preference, we may choose the semir-

ing S

FCSP

= h[0,1], max,min, 0, 1i and consider the

so-called fuzzy CSPs. As an example, consider the

following fuzzy CSP where V = {X,Y}, D = {a,b}

and C = {c

}. Soft constraint c

is deﬁned over

Y and associates preference 0.4 to a and to 0.7 to b.

Constraint c

, instead, is deﬁned over X and Y and

associates 0.9, 0.8, 0.7, 0.6 to, respectively, tuples

(X = a,Y = a), (X = a,Y = b), (X = b,Y = a) and

(X = b,Y = b).

Two main operations are deﬁned on soft con-

straints: combination, denoted with ⊗, and projec-

tion, denoted with ⇓. Combining two constraints

means building a new constraint involving all the

variables of the original ones, and which associates

to each tuple of domain values for such variables

a semiring element which is obtained by combining

(via ×) the elements associated by the original con-

straints to the appropriate subtuples. In the example

of the fuzzy CSP above, c

⊗ c

is a constraint on X

andY associating 0.4, 0.7, 0.4 and 0.6 to, respectively,

tuples (X = a,Y = a), (X = a,Y = b), (X = b,Y = a)

and (X = b,Y = b).

Projecting a constrainton a subset variables means

eliminating the other variables by associating to each

tuple over the remaining variables a semiring element

which is the sum (via +) of the elements associated by

the original constraint to all the extensions of this tu-

ple overthe eliminated variables. In the example, con-

straint c

⊗c

⇓

is a constraint deﬁned only overX,

which associates 0.7 to a and 0.6 to b.

To solve an SCSP, we just combine all constraints,

inducing an ordering over the set of all complete as-

signments. In the case of fuzzy CSPs, such and order-

ing is a total order with ties. In the example above, the

induced ordering has (X = a,Y = b) at the top with a

preference of 0.7, (X = b,Y = b) just below with 0.6

and (X = b,Y = a) and (X = b,Y = b) tied at the bot-

tom with 0.4. An optimal solution of an SCSP is then

a complete assignment with an undominated prefer-

ence. Finding an optimal solution in a set of soft con-

straints is an NP-hard problem.

Constraint propagation in SCSPs may be very

helpful in For some classes of constraints, con-

straint propagation is enough to solve the problem

(Dechter,2005). This is the case for tree-shaped fuzzy

CSPs, where directional arc-consistency (DAC), ap-

plied bottom-up on the tree shape of the problem, is

enough to make the search for an optimal solution

backtrack-free. DAC is also enough to compute the

preferences over the values of the root variable, in de-

pendence of the rest of the problem. That is, DAC is

equivalent to combining all constraints and projecting

over the root variable.

If we project the solution of a SCSP over a single

variable, we obtain a total order with ties over the val-

ues of that variable, where each value is associated to

the preference of the best solution of the SCSP hav-

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence

278

ing such variable instantiated to such a value. Given

an SCSP P and one of its variables v, we will denote

as top(v,P) the set of values of v that are assigned the

highest preference in such an ordering. We note that

such a preference coincides with the optimal prefer-

ence value of P. In our running example, if we con-

sider c

⊗ c

⇓

, the induced ordering over the val-

ues of the domain of X is a > b.

Voting Theory. In classical voting theory (Arrow

and amd K. Suzumura, 2002), given a set of candi-

dates C, a proﬁle is a sequence of Usually, such order-

ings are total orders, however several extensions have

been studied such as when the orderings are partial

orders or total orders with ties. Given a proﬁle, a vot-

ing rule, also known as social choice function, maps

it onto a single winning candidate. In this paper, we

will often use a terminology which is more familiar to

multi-agent settings, and we will therefore sometimes

call ”agents” the voters, ”solutions” the candidates,

and ”decision” or ”best solution” the winning candi-

date.

Some examples of widely used voting rules are:

Plurality: where each voter states who the preferred

candidate is, and the candidate who is preferred by the

largest number of voters wins; Borda: where given m

candidates, each voter gives a ranking of all candi-

dates and the i

ranked candidate scores m − i; the

candidate with the greatest sum of scores wins; Ap-

proval: where each voter approves between 1 and

m− 1 candidates on m total candidates; the candidate

with most votes of approval wins; Copeland: where

the winner is the candidate that wins the most pair-

wise competitions against all the other candidates.

The research on voting theory has mainly been

concerned with the deﬁnition of desirable properties

of voting systems. Among them, we recall:

• Condorcet-consistency: every other in pairwise

elections (namely, a Condorcet winner) exists,

that candidate is always elected; Condorcet win-

ners are unique and may not exist.

• Anonymity: When the results of an election are

the same even if it occurs a permutation on the

voters’ set.

• Neutrality: When it is anonymous w.r.t. the can-

didates.

• Monotonicity: voter improves his vote in favor of

this candidate, then the same candidate still wins.

• Consistency: If, when considering preferences of

2 disjoint sets of voters - who decide over the

same issues and have identical ﬁnal results - the

result obtained by a vote of the joint set of voters

is the same as the ones obtained by the disjoint set

of voters.

• Participation: If, given any proﬁle, and given a

new vote over a set of issues by a new voter, the

result obtained from the new proﬁle is equally or

more preferred by the new voter, who, thus, has

an incentive to participate. If, given a winner over

an election, there’s no candidate who is preferred

to the winner by all voters.

All the rules cited above are anonymous and

neutral,all but Cup are efﬁcient, only Cup and

Copeland are Condorcet consistent, and all but Cup

and Copeland are consistent and participative.

3 SEQUENTIAL PREFERENCE

AGGREGATION

In the fuzzy scenario, each variable assignment is

given a preference between 0 and 1, and assignments

with a higher preference value are more preferred.

Assume to have a set of agents, each one express-

ing his preferences over a common set of objects via

some soft constraints. as well as different preferences

over the variable domains. We will call a soft pro-

ﬁle the preference of a set of m agents, identiﬁed by

a triple (V,D, P): a set of variables V, a sequence D

of |V| domains, and a sequence P of m soft constraint

problems over variables in V with domains in D. A

fuzzy proﬁle is a soft proﬁle where the preferences of

the agents are fuzzy constraints.

The idea is to sequentially vote on each variable

via a voting rule. We do not restrict to use always the

same voting rule for all variables, so we will have a

sequence of as many voting rules as the variables.

Given a proﬁle (V,D,P), assume |V| = n, and con-

sider an ordering of such variables O = hv

,. .. ,v

}

and a sequence of voting rules R = hr

,. ..,r

i. The

sequential voting procedure we propose is a sequence

of n steps, where at each step i:

1. We ask all agents to report their preference order-

ing over the domain of variable v

. If we have

m agents, let us call such preference orderings

hpo

,. .. , po

2. We apply voting rule r

to this proﬁle, returning

a winning assignment for variable v

, say d

. If

there are ties in the result, the ﬁrst one following

a lexicographical order will be taken.

3. We add the constraint v

= d

to the preferences of

each agent.

After all n steps have been executed, the tuple

,. ..,d

i is reported as the chosen assignment for

the variables in V. We write Seq

O,R

(V,D, P) =

,. ..,d

MULTI-AGENT SOFT CONSTRAINT AGGREGATION - A Sequential Approach

279

This short description of the sequential voting pro-

cedure does not say what it means for an agent to re-

port their preference ordering over the domain of vari-

able v

. In general, since we do not make any assump-

tion on the voting rules r

, the agent needs to provide

the rule with a preference ordering over the whole do-

main of v

. Since this variable can be connected to

other parts of the agent’s soft constraint problem, in

order to report the correct preferences over the do-

main of v

, the agent needs to consider the inﬂuence

of the rest of the problem over v

. This means that,

in general, the agent needs to compute the projection

over v

of its whole soft constraint problem. This task

is in general difﬁcult, so it may require exponential

time to accomplish it, unless the class of constraint

problems used by the agent is tractable. This is for

example the case of tree-like shaped soft constraint

problems, which are polynomial to solve.

4 CONDORCET CONSISTENCY

It is natural to ask ourselves if the result returned

by the sequential voting procedure has some relation

with what is considered to be most preferred by the

agents.

A Condorcet winner (CW) is a candidate which

is preferred to any other candidate by a majority of

agents. Given a totally ordered proﬁle, as in clas-

sical voting theory, there can be zero or exactly one

Condorcet winner. In our context, since we may have

ties in the preference orderings of the agents, there

could be more than one Condorcet winner, since sev-

eral variable instantiations could be considered opti-

mal for a majority of agents.

First, we deﬁne the notion of sequential Con-

dorcet winner (SCW). Given an SCSP Q, we will

denote as Q|

,···,v

the problem obtained from

Q by ﬁxing variables v

,· ·· ,v

to the correspond-

ing values. Let P

denote the fuzzy constraint prob-

lem of agent i. Given a soft proﬁle (V,D, P) with

m agents and n variables, and an ordering O over V,

,. .. ,d

i is a SCW iff, for all j = 1,... ,n, |{i|d

∈

top(v

,...,v

−1=d

−1

)}| > m/2. In words, a se-

quential Condorcet winner is the combination of local

Condorcet winners.

If all the local rules are Condorcet consistent, the

sequential voting procedure returns a SCW by deﬁ-

nition. However, to conclude that Seq is Condorcet

consistent, we need to prove that SCW = CW. The

following results shows that a CW is always an SCW,

but unfortunately the opposite does not hold.

Theorem 1. Given a soft proﬁle (V, D,P) and an or-

dering O overV, if d is a CW for (V,D, P), it is a SCW

for (V,D,P). Thus, if Seq is Condorcet consistent, all

local voting rules are so.

If d = (d

,. .. ,d

) is a CW, then a majority of vot-

ers prefers it to all other candidates. Thus, at each step

i of the sequential voting procedure, the same major-

ity prefers d

to all other values in the domain of v

given the values already chosen.

The opposite does not hold in general, even if all

voting rules are Condorcet consistent.

Theorem 2. If all local voting rules are Condorcet

consistent, the sequential voting proceduremay be not

Condorcet consistent.

Consider a fuzzy proﬁle (V,D,P) where: V =

{X,Y}, D

= D

= {a,b} and there are 5 agents.

The fuzzy SCSPs of all agents havea single constraint

over {X,Y}. For two agents we have: def(X = a,Y =

b) = 0.9, def(X = b,Y = b) = 0.8, def(X = a,Y =

a) = 0.7, de f(X = b,Y = a) = 0.6; for one agent:

de f(X = a,Y = a) = 0.9, def(X = a,Y = b) = 0.8,

de f(X = b,Y = a) = 0.7, def (X = b,Y = b) = 0.6;

for the other two agents: def(X = b,Y = a) = 0.9,

de f(X = b,Y = b) = 0.8, def(X = a,Y = a) = 0.7,

de f(X = a,Y = b) = 0.6. When each agent solves the

problem and projects on variable X, for the ﬁrst two

agents we have pref

(a) = 0.9 and pref

(b) = 0.8;

for the third agent pref

(a) = 0.9 and pref

(b) =

0.7; and for the last two agents pref

(a) = 0.7 and

pref

(b) = 0.9. Thus, 3 over 5 agents agree that X =

a is optimal. Since the voting rule r

is Condorcet-

consistent, this value will be chosen for X. Given X =

a, the preferences of the agents for Y are: for the ﬁrst

two agents pref

(a) = 0.7 and pref

(b) = 0.9; for

the third agent pref

(a) = 0.9 and pref

(b) = 0.7;

for the last two agents pref

(a) = 0.7 and pref

(b) =

0.6. Thus Y = a will be chosen, since r

is Condorcet

consistent, and (X = a,Y = a) will be the SCW. How-

ever, (X = a,Y = a) is not a CW, since the majority

of the agents prefers (X = b,Y = b).

5 ANONYMITY, NEUTRALITY

AND CONSISTENCY

It is also important to make sure that a preference ag-

gregation system does not depend on the names or the

order of the agents. This corresponds to saying that

the rule is anonymous. In our setting, a permutation

of voter set corresponds, basically, to a permutation

of the soft constraint problems. It is easy to see that

if the sequential voting rule respects anonymity, then

also all the local voting rules do so, and, vice versa,

if all the local voting rules are anonymous, so is the

resulting sequential rule.

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence

280

Neutrality, on the other hand, is a property that

requires for a rule to be insensitive to permutations of

the candidates. This means that the result does not de-

pend on the names of the candidates, but only on their

position in the preference orderings. We note that the

candidates of the local voting rules are the values in

the variable domains, while the candidates of the se-

quential voting rule are the complete assignments to

all variables. While a permutation of the values in

the domains always corresponds to a permutation of

the variable assignments, not all of the permutations

of variable assignments can be obtained via permuta-

tions of domain values. Thus neutrality of the local

voting rules does not imply neutrality of the sequen-

tial voting rule, while neutrality of the sequential vot-

ing rule implies neutrality of each local voting rule.

As deﬁned above, a voting rule r is consistent if,

when considering two proﬁles P

and P

with disjoint

sets of voters, who vote over the same candidates,

such that r(P

) = r(P

), we have r(P

∪ P

) = r(P

Theorem 3. If all the local voting rules in R < The

opposite holds as well: if Seq

O,R

is consistent, then

all the local voting rules in R are consistent.

In fact, if all the local rules are consistent, at every

step i of the sequential procedure, applied to proﬁle

∪ P

, the result for variable v

is the same as the re-

sult in proﬁle P

(and also in proﬁle P

), so the overall

result (d

,. ..,d

) will be the same as the result ob-

tained by the sequential procedure in proﬁle P

and in

proﬁle P

. On the other hand, if one of the local rules

6 PARTICIPATION

Theorem 4. If the sequential voting procedure is par-

ticipative, then each local voting rule is so.

This means that there is a proﬁle over the val-

ues of variable v

, say p

, and an agent h with pref-

erence p

i,h

over the values of v

, such that agent h

strictly prefers r

) to r

∪ p

i,h

). for each agent

j = 1,. .. ,m, the preferences over the values of v

are

as in p

; all other unary constraints are the same for

all agents and associate preference 1 to exactly one

value per variable and 0 to all other variables; there

are no other constraints. Assume the SCSP P

agent h is deﬁned as follows: his preference over vari-

able v

is p

i,h

, all other unary constraints associate

preference 1 to exactly one value per variable and 0

to all other variables; there are no other constraints.

Since we have that Seq

O,R

(V,D, P) ↓ v

= r

) and

Seq

O,R

(V,D, (P ∪ P

)) ↓ v

= r

∪ p

i,h

), agent h

strictly prefers Seq

O,R

(V,D, P) to Seq

O,R

(V,D, (P ∪

)).

On the other hand, it is possible that all local vot-

ing rules are participative, but the sequential voting

procedure is not so.

Theorem 5. If all the local voting rules are partic-

ipative, the sequential voting procedure may not be

participative.

To see this, consider the proﬁle where V = {x,y},

D = ({a,b,c}, {a,b}), and P is a sequence of two

fuzzy SCSPs which coincide and contain a unary con-

straint on x (associating preference 1 to a, 0.8 to b,

and 0.6 to c), a binary constraint on x and y (as-

sociating preference 1 to (a,b), 0.9 to (a,a), 0.8 to

(b,a), 0.7 to (b,b), 0.6 to (c, a), and 0.5 to (c,b)),

and a unary constraint over y (associating preference

1 to both a and b). It is easy to see that this SC-

SPs are DAC. Assume also that variables are ordered

x ≺

y and that r

is the scoring rule with score vec-

tor (3,2,0) and r

is the majority rule. In this proﬁle,

Seq

O,R

(V,D, P) = (x = a,y = b). We now consider a

third voter, with a fuzzy SCSP with a unary constraint

on x (associating preference 0.8 to a, 1 to b, and 0.9 to

c), a binary constraint on x and y (associating prefer-

ence 0.8 to (a,b), 0.5 to (a,a), 0.7 to (b,a), 1 to (b,b),

0.6 to (c, a), and 0.5 to (c,b)), and a unary constraint

over y (associating preference 1 to both a and b). In

this new proﬁle P

′

, Seq

O,R

(V,D, P

′

) = (x = b,y = a).

However, the third voter prefers (x = a,y = b) to

(x = b,y = a). Thus the third voter would be better

off not participating to the sequential voting process.

7 EFFICIENCY

Theorem 6. If the sequential voting procedure is ef-

ﬁcient, then each local voting rule is so.

Let us assume that there is a local rule, say r

that

is not efﬁcient. This means that there is a proﬁle over

the values of variable v

, p

, such that r

) = d but

all agents prefer another value d

′

. Let us now con-

sider the soft proﬁle (V,D,P), where for each agent

j the preferences over the values of v

are as in p

;

all other unary constraints are the same for all agents

and associate preference 1 to exactly one value per

variable and 0 to all other variables and there are no

other constraints. We will have that Seq

O,R

(V,D, P) =

,. .. ,d

i−1

,d,...,d

). For each agent j his pref-

erence for Seq

O,R

(V,D, P) corresponds with his lo-

cal preference for d. Thus each agent will prefer

,. .. ,d

i−1

′

,. ..,d

) to Seq

O,R

(V,D, P).

On the other hand, it is possible that all local vot-

ing rules are efﬁcient, but the sequential voting proce-

dure is not so. However, if we add the condition that

there is a single optimal candidate for all agents, then

the sequential voting procedure is efﬁcient.

MULTI-AGENT SOFT CONSTRAINT AGGREGATION - A Sequential Approach

281

Theorem 7. If all the local voting rules are efﬁcient,

and there is a single candidate which is strictly pre-

ferred to all other candidates for all voters, then the

sequential voting procedure is efﬁcient.

In fact, if there is a single candidate, say d =

,. .. ,d

) that is optimal for all agents, then we have

that, after DAC, for each agent i and for each variable

j, top(v

,...,v

−1=d

−1

) = d

. Thus since each

rule is efﬁcient it will elect the value of d assigned to

its variable. Thus Seq

O,R

(V,D, P) = d.

We note that for proﬁles where there is a unique

candidate that is optimal for all agents, efﬁciency co-

incides with Condorcet consistency. Thus, given such

proﬁles, the Condorcet consistency of the local rules

implies that of the the sequential rule.

8 MONOTONICITY

As deﬁned above, a voting rule is monotonic if, when

a candidate wins, and one or more voters improve

their vote in favor of this candidate, then the same

candidate still wins.

Theorem 8. If the sequential voting procedure is

monotonic, then each local voting rule is so.

Let us assume that there is a local rule, say r

that is not monotonic. This means that there are

two proﬁles over the values of variable v

, say p

and p

′

, such that r

) = d

, p

′

is as p

except that

some agents have moved d

up in their orderings, and

′

) = d

′

6= d

Let us now consider the fuzzy proﬁle (V,D,P)

where: for each agent, the preferences over the values

of v

are as in p

; all other unary constraints are the

same for all agents and associate preference 1 to ex-

actly one value per variable and 0 to all other variables

and there are no other constraints. Assume the result

of applying the sequential rule is Seq

O,R

(V,D, P) =

,. .. ,d

Now let us consider the fuzzy proﬁle (V, D,P

′

)

where P

′

differs from P only on the preferences on

variable v

that, for each agent, are as in p

′

. We note

that in each agent’s SCSP, both in P and P

′

, there are

as many solutions as the values in the domain of v

and that their preference coincides with the one of the

corresponding value of v

. Thus P

′

differs from P only

on the preference of the only solution involving value

, i.e. Seq

O,R

(V,D, P), that has been moved up in the

ordering of some of the agents. However, the result

Seq

O,R

(V,D, P

′

) will be (d

,. .. ,d

′

,. .. ,d

). Thus

Seq is not monotonic.

Theorem 9. If each local rule is monotonic, so is the

sequential rule.

Let us assume that the sequential rule is not mono-

tonic. This means that there is at least one soft pro-

ﬁle (V,D,P) such that Seq

O,R

(V,D, P) = d and an-

other soft proﬁle (V,D,P

′

), where d has been moved

up in the preference orderings of some agents, but

Seq

O,R

(V,D, P

′

) = d

′

6= d. Let us denote with d

, resp.

′

, the value assigned to variable v

in d, resp. d

′

. We

note that there must be at least one value on which

they differ. For each SCSP in P

′

and for each variable

, d

has either improved w.r.t. d

′

or remained as in

P. Let v

be any of the variables such that d

6= d

′

Then r

is not monotonic.

The same results can be proven for strong mono-

tonicity, that is, all local voting rules are strongly

monotonic iff the sequential tule is so.

9 FUTURE WORK

We just considered a few properties of the sequential

voting procedure. We plan to study many others in or-

der to better characterize the result of the procedure in

terms of the preferences of the agents. We also plan to

developheuristics for an efﬁcient computing of such a

result even when the soft constraint problems are not

from a tractable class.

ACKNOWLEDGEMENTS

Research partially supported by the Italian MIUR

PRIN project 20089M932N: “Innovative and multi-

disciplinary approaches for constraint and preference

reasoning”.

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